A possible value of

Let $\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}=\theta$

$\sin 4 \theta=\frac{\sqrt{63}}{8}$

$\cos 4 \theta=\frac{1}{8}$

$2 \cos ^{2} 2 \theta-1=\frac{1}{8}$

$\cos ^{2} 2 \theta=\frac{9}{16}$

$\cos 2 \theta=\frac{3}{4}$

$2 \cos ^{2} \theta-1=\frac{3}{4}$

$\cos ^{2} \theta=\frac{7}{8}$

$\cos \theta=\frac{\sqrt{7}}{2 \sqrt{2}}$

$\tan \theta=\frac{1}{\sqrt{7}}$